Overview
The double integration method is a fundamental technique for evaluating beam deflection. It is based on the curvature definition and the flexure formula, which relate the bending moment \( M(x) \) to the deflection \( y(x) \) through the beam's elastic rigidity \( EI \).
Governing Equation
From the curvature relation and flexure formula:
$$ \frac{1}{\rho} = \frac{d^2 y}{dx^2} = \frac{M(x)}{EI} \tag{2.36} $$Rearranged:
$$ EI \cdot \frac{d^2 y}{dx^2} = M(x) $$Slope Equation
Integrating once with respect to \( x \), we obtain the slope \( \theta(x) = \frac{dy}{dx} \):
$$ EI \cdot \frac{dy}{dx} = \int M(x) \, dx + C_1 \tag{2.37} $$where \( C_1 \) is the first integration constant, determined from boundary conditions.
Deflection Equation
Integrating again gives the deflection \( y(x) \):
$$ EI \cdot y(x) = \int \left( \int M(x) \, dx + C_1 \right) dx + C_2 \tag{2.38} $$where \( C_2 \) is the second integration constant.
Boundary Conditions
The constants \( C_1 \) and \( C_2 \) are determined using support constraints such as:
- At a fixed support: \( y = 0 \), \( \theta = 0 \)
- At a simply supported end: \( y = 0 \)
- At a free end: slope or deflection may be non-zero
Practical Notes
- The method assumes continuous functions; break the beam into segments for varying loads.
- For cantilevers, origin is usually at the fixed end, often simplifying \( C_1 = C_2 = 0 \).
- Uniformly distributed loads (UDL) must be expressed over the full span, possibly using negative signs in \( M(x) \).
- Point moments are included in \( M(x) \) as constants (zero power of distance).
- For uniformly varying loads (UVL), use the intensity at the section being analyzed.